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/*
Copyright (C) 2012 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_mat.h"
slong
acb_mat_gauss_partial(acb_mat_t A, slong prec)
{
acb_t e;
acb_ptr * a;
slong j, m, n, r, rank, row, col, sign;
m = A->r;
n = A->c;
a = A->rows;
rank = row = col = 0;
sign = 1;
acb_init(e);
while (row < m && col < n)
{
r = acb_mat_find_pivot_partial(A, row, m, col);
if (r == -1)
{
break;
}
else if (r != row)
{
acb_mat_swap_rows(A, NULL, row, r);
sign *= -1;
}
rank++;
for (j = row + 1; j < m; j++)
{
acb_div(e, a[j] + col, a[row] + col, prec);
acb_neg(e, e);
_acb_vec_scalar_addmul(a[j] + col + 1, a[row] + col + 1, n - col - 1, e, prec);
}
row++;
col++;
}
acb_clear(e);
return rank * sign;
}
void
acb_vec_get_arf_2norm_squared_bound(arf_t s, acb_srcptr vec, slong len, slong prec)
{
slong i;
arf_t t;
arf_init(t);
arf_zero(s);
for (i = 0; i < len; i++)
{
arb_get_abs_ubound_arf(t, acb_realref(vec + i), prec);
arf_addmul(s, t, t, prec, ARF_RND_UP);
arb_get_abs_ubound_arf(t, acb_imagref(vec + i), prec);
arf_addmul(s, t, t, prec, ARF_RND_UP);
}
arf_clear(t);
}
void
acb_mat_det_lu_inplace(acb_t det, acb_mat_t A, slong prec)
{
slong i, n, sign, rank;
int is_real;
n = acb_mat_nrows(A);
rank = acb_mat_gauss_partial(A, prec);
sign = (rank < 0) ? -1 : 1;
rank = FLINT_ABS(rank);
_acb_mat_diag_prod(det, A, 0, rank, prec);
acb_mul_si(det, det, sign, prec);
/* bound unreduced part using Hadamard's inequality */
if (rank < n)
{
arf_t t;
arf_t d;
acb_t e;
arf_init(t);
arf_init(d);
acb_init(e);
arf_one(d);
is_real = acb_mat_is_real(A);
for (i = rank; i < n; i++)
{
acb_vec_get_arf_2norm_squared_bound(t, A->rows[i] + rank, n - rank, MAG_BITS);
arf_mul(d, d, t, MAG_BITS, ARF_RND_UP);
}
/* now d contains the absolute value of the determinant */
arf_sqrt(d, d, MAG_BITS, ARF_RND_UP);
/* multiply by disc with radius d */
if (is_real)
{
arb_add_error_arf(acb_realref(e), d);
}
else
{
arb_add_error_arf(acb_realref(e), d);
arb_add_error_arf(acb_imagref(e), d);
}
acb_mul(det, det, e, prec);
acb_clear(e);
arf_clear(d);
arf_clear(t);
}
}
void
acb_mat_det_lu(acb_t det, const acb_mat_t A, slong prec)
{
slong n;
n = acb_mat_nrows(A);
if (n == 0)
{
acb_one(det);
}
else
{
acb_mat_t T;
acb_mat_init(T, acb_mat_nrows(A), acb_mat_ncols(A));
acb_mat_set(T, A);
acb_mat_det_lu_inplace(det, T, prec);
acb_mat_clear(T);
}
}
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